Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & 1 & 2 & 1 1 & 1 & 0 & 2 1 & 0 & 1 & 0 0 & 2 & 1 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 0 & 1 & 2 & 1 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 0 \\ 0 & 2 & 1 & 1 \end{array}]$

Step 1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in column

1

$1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} + & - & + & - - & + & - & + + & - & + & - - & + & - & + \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Step 1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 2 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

0 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 2 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$0 | \begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$

Step 1.5

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$

Step 1.6

Multiply element

a_{21}

$a_{21}$ by its cofactor.

- 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$- 1 | \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$

Step 1.7

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} |$

Step 1.8

Multiply element

a_{31}

$a_{31}$ by its cofactor.

1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣

$1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} |$

Step 1.9

The minor for

a_{41}

$a_{41}$ is the determinant with row

4

$4$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 0 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{array} |$

Step 1.10

Multiply element

a_{41}

$a_{41}$ by its cofactor.

0 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 0 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣

$0 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{array} |$

Step 1.11

Add the terms together.

0 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 2 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ - 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 0 & 1 & 0 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 1 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 2 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 1 1 & 0 & 2 0 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣

$0 | \begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} | - 1 | \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{array} |$

Step 2

Multiply

$0$ by

$| \begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$ .

$0 - 1 | \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{array} |$

Step 3

Multiply

$0$ by

$| \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{array} |$ .

$0 - 1 | \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4

Evaluate

$| \begin{array}{c} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \end{array} |$ .

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Step 4.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$2$ by its cofactor and add.

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Step 4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 4.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 4.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$

Step 4.1.4

Multiply element

$a_{21}$ by its cofactor.

$0 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$

Step 4.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$

Step 4.1.6

Multiply element

$a_{22}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$

Step 4.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

$| \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$

Step 4.1.8

Multiply element

$a_{23}$ by its cofactor.

$0 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$

Step 4.1.9

Add the terms together.

$0 - 1 (0 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.2

Multiply

$0$ by

$| \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$ .

$0 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.3

Multiply

$0$ by

$| \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$ .

$0 - 1 (0 + 1 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} | + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.4

Evaluate

$| \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$ .

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Step 4.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - 1 (0 + 1 (1 \cdot 1 - 2 \cdot 1) + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.4.2

Simplify the determinant.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply

$1$ by

$1$ .

$0 - 1 (0 + 1 (1 - 2 \cdot 1) + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.4.2.1.2

Multiply

$- 2$ by

$1$ .

$0 - 1 (0 + 1 (1 - 2) + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.4.2.2

Subtract

$2$ from

$1$ .

$0 - 1 (0 + 1 \cdot - 1 + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.5

Simplify the determinant.

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Step 4.5.1

Multiply

$- 1$ by

$1$ .

$0 - 1 (0 - 1 + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.5.2

Subtract

$1$ from

$0$ .

$0 - 1 (- 1 + 0) + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 4.5.3

Add

$- 1$ and

$0$ .

$0 - 1 \cdot - 1 + 1 | \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} | + 0$

Step 5

Evaluate

$| \begin{array}{c} 1 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 1 \end{array} |$ .

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Step 5.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$2$ by its cofactor and add.

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Step 5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 5.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$

Step 5.1.4

Multiply element

$a_{21}$ by its cofactor.

$- 1 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$

Step 5.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$

Step 5.1.6

Multiply element

$a_{22}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$

Step 5.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

$| \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$

Step 5.1.8

Multiply element

$a_{23}$ by its cofactor.

$- 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$

Step 5.1.9

Add the terms together.

$0 - 1 \cdot - 1 + 1 (- 1 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} | - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.2

Multiply

$0$ by

$| \begin{array}{c} 1 & 1 \\ 2 & 1 \end{array} |$ .

$0 - 1 \cdot - 1 + 1 (- 1 | \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} | + 0 - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.3

Evaluate

$| \begin{array}{c} 2 & 1 \\ 1 & 1 \end{array} |$ .

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Step 5.3.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - 1 \cdot - 1 + 1 (- 1 (2 \cdot 1 - 1 \cdot 1) + 0 - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.3.2

Simplify the determinant.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply

$2$ by

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 (2 - 1 \cdot 1) + 0 - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.3.2.1.2

Multiply

$- 1$ by

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 (2 - 1) + 0 - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.3.2.2

Subtract

$1$ from

$2$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 | \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |) + 0$

Step 5.4

Evaluate

$| \begin{array}{c} 1 & 2 \\ 2 & 1 \end{array} |$ .

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Step 5.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 (1 \cdot 1 - 2 \cdot 2)) + 0$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Multiply

$1$ by

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 (1 - 2 \cdot 2)) + 0$

Step 5.4.2.1.2

Multiply

$- 2$ by

$2$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 (1 - 4)) + 0$

Step 5.4.2.2

Subtract

$4$ from

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 \cdot 1 + 0 - 2 \cdot - 3) + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply

$- 1$ by

$1$ .

$0 - 1 \cdot - 1 + 1 (- 1 + 0 - 2 \cdot - 3) + 0$

Step 5.5.1.2

Multiply

$- 2$ by

$- 3$ .

$0 - 1 \cdot - 1 + 1 (- 1 + 0 + 6) + 0$

Step 5.5.2

Add

$- 1$ and

$0$ .

$0 - 1 \cdot - 1 + 1 (- 1 + 6) + 0$

Step 5.5.3

Add

$- 1$ and

$6$ .

$0 - 1 \cdot - 1 + 1 \cdot 5 + 0$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply

$- 1$ by

$- 1$ .

$0 + 1 + 1 \cdot 5 + 0$

Step 6.1.2

Multiply

$5$ by

$1$ .

$0 + 1 + 5 + 0$

Step 6.2

Add

$0$ and

$1$ .

$1 + 5 + 0$

Step 6.3

Add

$1$ and

$5$ .

$6 + 0$

Step 6.4

Add

$6$ and

$0$ .

$6$

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